Imbo Sim

Myoungnyoun Kim, Imbo Sim

The order of convergence of the Monte Carlo method is 1/2 which means that we need quadruple samples to decrease the error in half in the numerical simulation. Multilevel Monte Carlo methods reach the same order of error by spending less computational time than the Monte Carlo method. To reduce the computational complexity further, we introduce a projected multilevel Monte Carlo method. Numerical experiments validate our theoretical results.

Ohsung Kwon, Imbo Sim

We present the numerical dispersion effects in solving the convected Helmholtz equation by the conforming and nonconforming quadrilateral finite elements. Particularly, we evaluate the dispersion relations for the numerical schemes. The dispersive behaviors are analyzed by focusing on the Mach number and the angular frequency. Numerical experiments are conducted to verify the relations between the numerical dispersions and the computational errors.

Myoungnyoun Kim, Imbo Sim

We present a reduced basis approach to solve the convected Helmholtz equation with several physical parameters. Physical parameters characterize the aeroacoustic wave propagation in terms of the wave and Mach numbers. We compute solutions for various combinations of parameters and spend a lot of time to figure out the desired set of parameters. The reduced basis method saves the computational effort by using the Galerkin projection, a posteriori error estimator, and greedy algorithm. Here, we propose an efficient a posteriori error estimator based on the primal norm. Numerical experiments demonstrate the good performance and effectivity of the proposed error estimator.

Seungwoo Lee, Do Y. Kwak, Imbo Sim

We consider the approximation of eigenvalue problems for elasticity equations with interface. This kind of problems can be efficiently discretized by using immersed finite element method (IFEM) based on Crouzeix-Raviart P1-nonconforming element. The stability and the optimal convergence of IFEM for solving eigenvalue problems with interface are proved by adapting spectral analysis methods for the classical eigenvalue problem. Numerical experiments demonstrate our theoretical results.

Seungwoo Lee, Do Y. Kwak, Imbo Sim

We consider the approximation of elliptic eigenvalue problem with an immersed interface. The main aim of this paper is to prove the stability and convergence of an immersed finite element method (IFEM) for eigenvalues using Crouzeix-Raviart $P_1$-nonconforming approximation. We show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem. We analyze the IFEM for elliptic eigenvalue problem with an immersed interface and derive the optimal convergence of eigenvalues. Numerical experiments demonstrate our theoretical results.

Marcus J. Grote, Imbo Sim

In the last decade, the perfectly matched layer (PML) approach has proved a flexible and accurate method for the simulation of waves in unbounded media. Most PML formulations, however, usually require wave equations stated in their standard second-order form to be reformulated as first-order systems, thereby introducing many additional unknowns. To circumvent this cumbersome and somewhat expensive step, we instead propose a simple PML formulation directly for the wave equation in its second-order form. Inside the absorbing layer, our formulation requires only two auxiliary variables in two space dimensions and four auxiliary variables in three space dimensions; hence it is cheap to implement. Since our formulation requires no higher derivatives, it is also easily coupled with standard finite difference or finite element methods. Strong stability is proved while numerical examples in two and three space dimensions illustrate the accuracy and long time stability of our PML formulation.